Chapter 7
Transformations
Chapter Objectives
Identify different types of transformations
Define isometry
Identify reflection
Identify rotations
Identify translations
Describe composition transformations
Lesson 7.1
Rigid Motion in a Plane
Lesson 7.1 Objectives
Identify basic rigid transformations
Define isometry
Definition of Transformation
A transformation is any operation that maps, or moves, an object to another location or orientation.
Transformation Terms
When performing a transformation, the original figure is called the pre-image.
The new figure is called the image.
Many transformations involve labels
The image is named after the pre-image, by adding a prime symbol (apostrophe)
A A’ A’’
We say it as “A prime”
Types of Transformations
Types Reflection Rotation Translation
Characteristics
Orientation
Pictures
Flips object over line of reflection
Turns object using a fixed point as center or rotation
Slides object through a plane
Order in which object is drawn is
reversed
Stays same just tilted
Stays same and stays upright
Definition of Isometry
An isometry is a transformation that preserves length.
Isometry also preserve angle measures, parallel lines, and distances between points.
If you look at the meaning of the two parts of the word, iso- means same, and metry- means meter or measure.
So simply stated, isometry preserves size.
Any transformation that is an isometry is called a Rigid Transformation.
Homework 7.1
1-33, 36-39p399-401
In Class – 9, 13, 27, 33
Due Tomorrow
Lesson 7.2
Reflections
Lesson 7.2 Objectives
Utilize reflections in a plane
Define line symmetry
Derive formulas for specific reflections in the plane
Reflections
A transformation that uses a line like a mirror is called a reflection.
The line that acts like a mirror is called the line of reflection.
When you talk of a reflection, you must include your line of reflection
A reflection in a line m is a transformation that maps every point P in the plane to a point P’, so that the following is true If P is not on line m, then m is the perpendicular bisector of PP’. If P is on line m, then P=P’.
Theorem 7.1:Reflection Theorem
A reflection is an isometry.That means a reflection does not change
the shape or size of an object!
m
Line of SymmetryA figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in a line.What that means is a line can be drawn through an object so that each side reflects onto itself.There can be more than one line of symmetry, in fact a circle has infinitely many around.
Homework 7.2a
1-11, 22-29p407-408
In Class – 7, 23
Due Tomorrow
Reflection Formula
There is a formula to all reflections.
It depends on which type of a line are you reflecting in. vertical horizontal y = x
Vertical:y-axisx = a
( -x + 2a , y)
Horizontal:
x-axisy = a
( x , -y + 2a)
y = x
( y , x)
( x , y)
Homework 7.2b
12-14, 18-21, 50-51p407-410
In Class – 19
Due Tomorrow
Lesson 7.3
Rotations
Lesson 7.3 Objectives
Utilize a rotation in a plane
Define rotational symmetry
Observe any patterns for rotations about the origin
Definitions of Rotations
A rotation is a transformation in which a figure is turned about a fixed point.
The fixed point is called the center of rotation.
The amount that the object is turned is the angle of rotation.
A clockwise rotation will have a negative measurement.
A counterclockwise rotation will have a positive measurement.
Q
clockwiseornegative (-)
Theorem 7.2:Rotation Theorem
A rotation is an isometry.
A
B
A’
B’
P
Rotational Symmetry
A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less. A square has rotational symmetry because it maps onto
itself with a 90° rotation, which is less than 180°. A rectangle has rotational symmetry because it maps onto
itself with a 180° rotation.
Homework 7.3a
1-19p416
In Class – 6, 11, 13
Due Tomorrow
Rotating About the Origin
Rotating about the origin in 90o turns is like reflecting in the line y = x and in an axis at the same time!
So that means to switch the positions of x and y. (x,y) (y,x)
Then the original x-value changes sign, no matter where it is flipped to.
So overall the transformation can be described by (x,y) (-y,x)
Every time you 90o you repeat the process. So going 180o means you do the process twice!
Theorem 7.3:Angle of Rotation Theorem
The angle of rotation is twice as big as the angle of intersection. But the intersection must be the center of rotation. And the angle of intersection must be acute or right only.
P
A
B
A’
B’
m
k
x
2x
Homework 7.3b
25-35, 45-50, 54p417-419
In Class – 25, 35
Due Tomorrow
Quiz WednesdayLessons 7.1-7.4
Lesson 7.4
Translations
and
Vectors
Lesson 7.4
Define a translation
Identify a translation in a plane
Use vectors to describe a translation
Identify vector notation
Translation Definition
A translation is a transformation that maps an object by shifting or sliding the object and all of its parts in a straight light.
A translation must also move the entire object the same distance.
Theorem 7.4:Translation Theorem
A translation is an isometry.
Theorem 7.5:Distance of Translation Theorem
The distance of the translation is twice the distance between the reflecting lines.
P
Q
P’
Q’
P’’
Q’’
x
2x
k m
Coordinate form
Every translation has a horizontal movement and a vertical movement.
A translation can be described in coordinate notation. (x,y) (x+a , y+b)Which tells you to move a units horizontal
and b units vertical.
a units to the right
b units upP
Q
Vectors
Another way to describe a translation is to use a vector.
A vector is a quantity that shows both direction and magnitude, or size. It is represented by an arrow pointing from pre-
image to image. The starting point at the pre-image is called the initial
point. The ending point at the image is called the terminal
point.
Component Form of Vectors
Component form of a vector is a way of combining the individual movements of a vector into a more simple form. <x , y>
Naming a vector is the same as naming a ray. PQ
x units to the right
y units upP
Q
Use of Vectors
Adding/subtracting vectors Add/subtract x values and then add y values
<2 , 6> + <3 , -4> <5 , 2>
Distributive property of vectors Multiply each component by the constant
5<3 , -4> <15 , -20>
Length of vector Pythagorean Theorem
x2 + y2 = lenght2
Direction of vector Inverse tangent
tan-1 (y/x)
Homework 7.4
1-30, 44-47p425-427
In Class – 3,7,17,25,45
Due Tomorrow
Quiz TomorrowLessons 7.1-7.4
Lesson 7.5
Glide Reflections
and
Compositions
Lesson 7.5 Objectives
Identify a glide reflection in a plane
Represent transformations as compositions of simpler transformations
Glide Reflection Definition
A glide reflection is a transformation in which a reflection and a translation are performed one after another.
The translation must be parallel to the line of reflection. As long as this is true, then the order in which the
transformation is performed does not matter!
Compositions of Transformations
When two or more transformations are combined to produce a single transformation, the result is called a composition.So a glide reflection is a composition.
The order of compositions is important!A rotation 90o CCW followed by a reflection
in the y-axis yields a different result when performed in a different order.
Theorem 7.6:Composition Theorem
The composition of two (or more) isometries is an isometry.
Homework 7.5
1-8, 9-21, 23-24, 26-30skip 16, 28p433-435
In Class – 9,13,19
Due Tomorrow
Lesson 7.6
Frieze Patterns
Lesson 7.6 Objectives
Identify a frieze pattern by type
Visualize the different compositions of transformations
Frieze Patterns
A frieze pattern is a pattern that extends to the left or right in such a way that the pattern can be mapped onto itself by a horizontal translation.Also called a border pattern.
Classifying Frieze Patterns
The horizontal translation is the minimum that must exist.However, there are other transformations that can occur. And they can occur more than once.
Type Abbreviation Description
Translation T Horizontal translation left or right
180o Rotation R 180o Rotation CW or CCW
Reflection inHorizontal Line
HReflection either up or down
in a horizontal line
Reflection inVertical Line
V Reflection either left or rightin a vertical line
HorizontalGlide Reflection
GHorizontal translation with
reflection in a horizontal line
Examples
TR TG
TV
THG
TRVGTRHVG
Homework 7.6
2-23p440-441
In Class – 9,13,17,21
Due Tomorrow
Quiz TuesdayLessons – 7.5-7.6
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